state observers and the kalman filter - unitrentobursi/download/phd_lectures_02july...luenberger...

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State Observers and the Kalman filter Prof. Oreste S. Bursi University of Trento

Modelling and Control of Dynamic Systems

July 6, 2012

Feedback System

State variable feedback system:

Control feedback law:u = -Kx

The control law requires that

all state variables are known July 6, 2012

• x- states; y- measurements; u- input;• A- State transition Matrix; • B - Input matrix; • C - Measurement matrix• r- reference signal - desired state

variables • K- state feedback gain matrix

The closed loop system is:

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Feedback System

r - reference signal

K- state feedback gain matrix

x- states

C - measurement matrixB - input matrix

A- state transition Matrix

y- measurementsu- input

Feedback System

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• Some of the state variables may not be measurable.

• The cost of many transducers may be too high.

Availability of State Variables

State Observer

Therefore,- we discard the assumption that all the state variables are measurable.

-we need something that gives an estimate of the state variables x for feedback, from the measurement of the controlled output y and from the input u.

This something is termed as the State Observer.

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State Observer

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Types:

• Full-order state observer- Estimates all of the state variables

• Reduced order state observer- Estimates some of the state variables

Full- order State Observer

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y - actual output

x - actual state variables

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Full- order State ObserverLuenberger Observer (1964)


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Luenberger Observer (1964)

Feedback on the error

between real measurement

and estimated

one

The Luenberger full order

state observer


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Since

The dynamic behaviour of the error vector depends upon theeigenvalues of

As with any measurement system, these eigenvalues should allowthe observer transient response to be more rapid than the systemitself (typically a factor of 5), unless a filtering effect is required.


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The problem of observer design is essentially the same as theregulator pole placement problem, and similar techniques may beused.

The following techniques may be used.

• Direct comparison method• Observable canonical form method• Ackermann’s formula

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• Direct comparison method

If the desired locations of the closed-loop poles (eigenvalues) of the observer are:

Then the characteristic polynomial of (A-KeC) is:

Full- order State ObserverSolving the eigenvalues of (A-KeC)

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• Observable canonical form method

For a generalized transfer function shown above, the observable canonical form of the state equation may be written as


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• Ackermann’s Formula- only applicable to systems where u(t) and y(t) are scalar quantities.

The observer gain matrix may be can be calculated as follows

where is defined as



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Effect on a closed-loop system

Closed-loop control

system with full order

state feedback

The equations of this closed-loop system are


July 6, 2012

Effect on a closed-loop system

Control is implemented using observed state variables,

The equations of this closed-loop system are

Error equation, , then

Combining above equations:

and remembering that the closed loop dynamic of the observer is:

The system characteristic equation isThe system characteristic equation shows that the desired closed-loop poles

for the control system are not changed by the introduction of the state observer.

Reduced- order State Observer

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A reduced-order state observer does not measure all the state variables.

If the state vector is of nth order and the measured output vector is of mth order, then it is only necessary to design an (n - m)th order state observer.

Consider the case of the measurement of a single state variable . The output equation is therefore

Reduced- order State Observer

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In the above equation

The reduced-order observer gain matrix can be obtained using methods discussed earlier in case of full-order observer.

The Kalman Filter

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State estimation is the process of extracting a best estimate of a variablefrom a number of measurements that contain noise.

The classical problem of obtaining a best estimate of a signal bycombining two noisy continuous measurements of the same signal wasfirst solved by Weiner (1949).

His solution required that both the signal and noise be modelled as randomprocess with known statistical properties.

Kalman and Bucy (1961) extended this work by designing a stateestimation process based upon an optimal minimum variance filter,generally referred to as a Kalman filter.

Evolution

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• In the design of state observers, it was assumed that the measurements y = Cx were noise free;

• However, this is not usually the case: also measurements are corrupted by noise.

• The observed state vector may also be contaminated with noise.• Hence, a filter is required to remove the effect of noise.• The Kalman filter is used for this purpose: noisy data in - less

noisy data out;• But delay is the price for filtering;• Need of the process model for describing how states change

over time.

The Kalman FilterIntroduction

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The Kalman FilterIntroduction

KF operates by:• Predicting the new state and its uncertainty;• Correcting with the new measurements.

The Kalman Filter

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Single Variable Estimation Problem

The Kalman Filter

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If P is the variance of the weighted mean,(2)

The optimal value of K is the one that yields the minimum variance, i.e.

Substitution of equation (1) into (2) gives

The Kalman Filter

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Substitution of equation (3) into (1) provides

Equation (4) is illustrated in the next Figure.

The Kalman Filter

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The Kalman FilterModel and initialization

Process and measurement model:

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The Kalman FilterPrediction-Correction

Gain in the measurement space

transition uncertainty Actualmeasurement

Predictedmeasurement

The Kalman Filter

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Multivariable Estimation Problem

Consider a plant that is subject to a Gaussian sequence of disturbances w(kT)with disturbance transition matrix Cu(T). Measurements z(k + 1)T contain a Gaussian noise sequence v(k + 1)T as shown in the following Figure.

Measurements vector

State time evolution

The Kalman Filter

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The term (k+1/k) means data at time k+1 based on information available at time k.

The Kalman Filter

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The vector measurement is given by:

(3)

The Kalman gain matrix K is obtained from a set of recursive equations that commence from some initial covariance matrix P(k|k):

(4)(5)(6)

where:z(k+1)T is the measurement vector;C(T) is the measurement matrix;v(k+1)T is a Gaussian noise sequence;Cd(T) is the disturbance transition matrix;Q is the disturbance noise covariance matrix;R is the measurement noise covariance matrix.

The Kalman Filter

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prediction

correction

The recursive processcontinues by substitutingthe covariance matrixP(k + 1/k + 1) computed

in equation (6) back into(4) as P(k/k) untilK(k+1) settles to a steadyvalue.

Reference

July 6, 2012

• Burns R. S., Advanced Control Engineering, 2001, Butterworth-Heinemann, Jordan Hill, Oxford OX2 8DP.

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